Collective effects in the energy loss of large hydrogen clusters

Collective effects in the energy loss of large hydrogen clusters

F. Javier Pe´rez-Pe´rez and Isabel Abril

Departament de Fı´sica Aplicada, Universitat d’Alacant, Apartat 99, E-03080 Alacant, Spain

Rafael Garcia-Molina

Departamento de Fı´sica, Universidad de Murcia, Apartado 4021, E-30080 Murcia, Spain

Ne´stor R. Arista

Instituto Balseiro, Centro Ato´mico Bariloche, RA-8400 Bariloche, Argentina

~Received 15 May 1996!

The energy loss of large molecular-hydrogen clusters incident on aluminum and amorphous carbon targets is analyzed as a function of the cluster size and velocity, using a dielectric formalism to describe the electronic interactions between the projectile and the target. The dependence of the energy loss with the cluster size predicts a marked different behavior for aluminum or amorphous carbon. The energy loss of H₂clusters in aluminum indicates that for each cluster velocity there is a ‘‘resonant’’ cluster size for which the intermolecu- lar contribution to the energy loss is maximum. On the other hand, the intermolecular contribution to the energy loss of H₂clusters in amorphous carbon saturates with the cluster size. The origin of this behavior lies in the low-energy plasmon present in the energy loss function of amorphous carbon. We also discuss the consequences of a proper description of the dielectric properties of the target.@S1050-2947~96!08810-5#

PACS number~s!: 34.50.Bw, 36.40.2c

I. INTRODUCTION

The interaction of atomic particles with condensed matter has been studied in depth and profusely from the beginning of this century. Nonetheless, during recent years a new inter- est has grown in the interaction of polyatomic projectiles with matter @1#, due to the possibility of producing clusters made by a large number of atoms, and at relatively high kinetic energies, which now extends up to the MeV/atom range.

The interest in cluster beams interacting with solids and plasmas was motivated by the fact that it provides additional information about a variety of atomic-collision phenomena and also due to several applications ~like implantation, sur- face analysis, sputtering, determination of geometrical struc- tures of polyatomic ions, inertial fusion drivers, etc.! @1#. In a pioneering work, Brandt et al. @2# measured and calculated the energy loss of H₂¹ and H₃¹ beams in thin foils and compared the results with the stopping power corresponding to proton beams at the same velocity. The conclusion of this paper was that the energy loss of the cluster shows important differences, called vicinage effects, with respect to the en- ergy loss of its constituents considered separately. The origin of this effect lies in the interferences produced by the elec- tronic excitations of the target due to the correlated motion of the particles that form the cluster.

At present, the study of the basic interactions between fast molecular-ion beams interacting with solids has received re- newed interest, both theoretically @3–8# and experimentally

@3,9–11#. However, works related to the interaction of large clusters with matter are scarcer, although we should quote recent experimental work concerning the interaction of large hydrogen and carbon clusters with solids @12–16#. The ex- perimental stopping power ratio of hydrogen clusters in car-

bon foils and its dependence on both cluster velocity and size was presented in Ref. @15#, for cluster energies in the range of keV/amu and cluster sizes up to 25 atoms. At low energies they@15# found that the ratio between the total energy loss of a H_n¹ cluster and the energy loss of n individual protons at the same velocity is less than unity, but it becomes larger than unity at high energies, in agreement with earlier obser- vations using H₂¹ions@17,18#.

Saturation effects in the cluster energy loss due to the cluster size were observed at least for the lower cluster en- ergies @15#. Recently the stopping power of carbon clusters and fullerene ions has been measured in carbon foils@16#, in the energy range of MeV per atom.

A theoretical treatment of the electronic stopping power of small clusters was first provided within the dielectric for- malism @19#. In Ref. @20# the energy loss of slow hydrogen and water clusters was analyzed for an electron gas at finite temperature. The effects of the cluster structure and velocity on the stopping power were also evaluated @21,22# for swift and large hydrogen clusters in carbon and aluminum targets.

A Monte Carlo code@23# was developed to treat the different processes involved in the interaction of hydrogen clusters, with energies in the keV/amu range, in thin carbon foils.

Other recent developments include energy-loss studies of fullerenes in a Lindhard gas@24#, and the cluster-size depen- dence of the stopping power for hydrogen clusters in silicon

@25#. Besides, the analysis of collective effects in the energy loss of large clusters in plasmas @26–28# or solids @29# has also received recent attention due to the possibility of obtain- ing higher densities of deposited energy.

Our aim in this work is to evaluate how the stopping power of large molecular-hydrogen clusters depends on the cluster size and velocity. We also discuss the role played by the target nature on the above mentioned study. In particular,

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1050-2947/96/54~5!/4145~8!/$10.00 4145 © 1996 The American Physical Society

we analyze the case of aluminum and amorphous carbon, because they are commonly used as targets, and also because they have rather different dielectric properties.

The paper is organized as follows. Section II introduces the dielectric formalism used to obtain the cluster energy loss, and the model adopted to describe the dielectric prop- erties of the target. In Sec. III we discuss the dependence of the cluster stopping power with the cluster size and velocity;

the different results obtained for aluminum and amorphous carbon targets will be also analyzed. Finally, the conclusions are presented in Sec. IV.

II. DIELECTRIC DESCRIPTION OF CLUSTER ENERGY LOSS

A. Stopping power

When swift clusters impinge on a solid their binding elec- trons are stripped off after traversing the first atomic layers

@2#, forming then a cluster of atomic ions. These ions interact with the target and also among them~under the influence of Coulomb forces!. However, since two time-scale regimes can be distinguished in these interactions ~the electronic time-scale, ;10²¹⁷–10²¹⁶ s, and the Coulomb explosion time-scale,;10²¹⁵–10²¹⁴s!, it is possible to evaluate sepa- rately both processes.

It should be noted that swift clusters inside solids are exploding projectiles in which each ion suffers change- exchange and multiple scattering interactions. Then, equilib- rium times needed for the loss of the projectile electrons and for the proton charge equilibrium may be in some cases not negliglible when compared with Coulomb explosion times.

However, the complexity of the whole problem makes a de- tailed study unpractical, therefore we will not consider these effects in our work.

In the context of swift cluster-beams bombarding a solid, when the cluster velocity v is larger than Fermi velocity, v_F, it loses energy due mainly to the interaction with the target electrons. Then, in what follows we will consider only the electronic stopping power of swift clusters in solids, ne- glecting the contribution of nuclear stopping power.

The correlated motion of the cluster partners produces the so-called vicinage effects@2#, which, up to first order pertur- bation theory, are satisfactorily evaluated within the dielec- tric formalism of the stopping power@19#. Within this frame- work, each cluster constituent is treated as a pointlike charge, and the target is modeled as an isotropic and homogeneous electron gas, whose response to an external perturbation is characterized completely by its dielectric function e^(k,v^), where k and v are, respectively, the momentum and the energy transferred to electronic excitations in the medium.

We will use atomic units throughout this paper.

When a charge penetrates into the solid it induces an elec- tric field, which acts back on the projectile and causes it to lose kinetic energy. If we consider a cluster with velocity vW, composed by N ions with relative positions rW_{i j}, then a general expression for the cluster stopping power, S_cl, is given by @19#

S_cl5Sp

F

⁽

⁽

G

where Z_i* is the effective charge of each individual particle in the cluster. This effective charge is an average value that reflects the balance between the ionic electron-capture and loss processes. We assume that the effective charges Z_i*are all equal and identical for atomic and cluster ions, i.e., Z_i*_5Z*. r_{i j} are the distances between ions i and j in the cluster and ui j are the angles made by each relative vector position rW_{i j} and the velocityvW of the cluster. The stopping power of a single proton, S_p, is expressed as

S_p5 1

2p^2v

E

F

G

The vicinage effects are explicitly taken into account through the interference function I(r,u), which can be written as@20#

I~r,u!5 1

2p^2vSp

E

F

G

~3!

A random distribution of the particles in the cluster is suitable for large clusters @22#. After doing an angular aver- age in Eq. ~3!, and introducing the cluster pair distribution function g_cl(r), which accounts for the correlated positions of the particles in the cluster@20,22#, we obtain the following expression for the cluster stopping power:

S_cl5NZ*²

H

^E

^E

3

E

dvv^Im

F

G J

with n¯ being the average nuclear density of the cluster; in our case we take n¯56.27310²³ a.u. for hydrogenic clusters, according to Ref.@30#.

B. Cluster geometry

The cluster geometry is described through the pair distri- bution function g_cl(r), which gives the probability that two particles in the cluster are at a relative distance r. This dis- tribution satisfies the normalization condition N511n¯*d³rg_cl(r), where the integration extends to the volume of the cluster. Since we are interested in large clus- ters of molecular hydrogen, ~H2)_n, we separate g_cl(r) in two contributions: g_cl(r)5gintra(r)1ginter(r). The intramo- lecular term, g_intra(r), describes the almost fixed separation of the protons in each H₂molecule, while the intermolecular term, g_inter(r), provides the spatial distribution of the rest of molecules in the cluster.

The structure of the H₂ molecule was represented by a d distribution, g_intra(r)5(4p^rmol

2 ¯)n ²¹d^(r2rmol), where r_mol51.40 a.u. is the corresponding internuclear distance

@30#. The intermolecular pair correlation function ginter(r) was modeled by a random distribution of intermolecular dis- tances in a spherical cluster with an exclusion volume around each molecule, due to the repulsive interaction between H₂ molecules at close distances; this exclusion distance is taken r_ex53.59 a.u. @30#. Finally, the finite spherical geometry of

the cluster was also incorporated in g_cl(r), according to the procedure presented in Ref.@22#.

We define the reduced stopping power as the stopping power per particle, normalized to the square of the effective charge, namely, S_cl/(NZ*²). It can be separated into contri- butions due to each proton not being part of the cluster, S_p, the interference effects produced by the protons of the same molecule, S_intra, and the interference generated by pro- tons from different molecules, S_inter. Then we can write

S_cl

NZ*^{25Sp1Sintra1Sinter. ~5!}

C. Dielectric description of the target

The cluster energy loss@see Eq. ~4!# depends on the target properties through the energy loss function~ELF, hereafter!, Im@21/e^(k,v⁾#, which should be specified for each of the targets considered in this work: aluminum and amorphous carbon.

We model the ELF using the dielectric function intro- duced by Mermin@31#,eM, which is a generalization of the Lindhard dielectric function @32# in the relaxation time ap- proximation. With this formalism, the finite lifetime for the plasmon oscillations and their decay into individual excita- tions are properly taken into account.

The procedure we have used to model the ELF has been described in detail elsewhere@33,34#, where we have shown that it gives satisfactory estimations of proton stopping pow- ers @33#, as well as an accurate description of the k andv dependence of the ELF@34#. In brief, we construct the ELF by fitting the experimental ELF, usually available at zero wave number, by means of a sum of Mermin-type ELF’s.

The number of terms depends roughly on the number of peaks in the ELF:

Im

F

G

⁽

F

G

The parametersvpl,i, andgiare related to the energy posi- tion and width of the peaks in the ELF, respectively, and the coefficients A_i are chosen in such a manner that the f -sum rule

N_eff~v!5 1 2p²r

E

v

dv8v8^Im

F

G

will be satisfied~this gives the effective number of electrons per atom that participate in the excitations up to energyv^).

In the above expression, r is the target atomic density.

Using the experimental ELF at k50 for aluminum @35#

and for amorphous carbon @36# we obtain the set of Ai, vpl,i, and gi values reported in Table I. In Fig. 1 we have depicted the experimental and fitted ELF’s of aluminum

@Fig. 1~a!# and amorphous carbon @Fig. 1~b!#. The aluminum energy loss spectrum presents a very well defined peak at the energy vpl'0.55 a.u. in the large wavelength limit, as can be seen in Fig. 1~a!. The ELF of amorphous carbon clearly shows two peaks, which describe the collective excitations of p ^{and (}p1s) electrons in carbon@36,37#; experimental evi-

dence @38# also indicates the presence of large damping of the plasma oscillations in carbon.

In some previous studies @3,21,39# the carbon ELF has been described with a single plasmon energy vpl50.792 a.u., obtained according to the relation vpl

2 54pN, where N is the free electron density of carbon, and with a damping g50.58 a.u. To illustrate the need for a more accurate target FIG. 1. ~a! Energy loss function of aluminum. Dashed line, ex- perimental data@35#; solid line, fitted ELF using Eq. ~6!. ~b! Energy loss function of amorphous carbon. Dashed line, experimental data

@36#; thick solid line, fitted ELF using Eq. ~6! using two Mermin- type ELF’s; thin solid line, fitted ELF using a single Mermin ELF with the plasmon energy obtained from the amorphous carbon den- sity.

description, in the next section we will compare the predic- tions of the cluster stopping powers obtained using the single plasmon ELF or the sum of Mermin-type ELF’s.

III. RESULTS AND DISCUSSION

We will now consider the dependence, on cluster size and velocity, of the stopping power of~H2)_n clusters on alumi- num and amorphous carbon. We compare the prediction for these two materials that present different dielectric re- sponses: the aluminum ELF, characterized by one very-well- defined plasmon with small damping @35#, and the amor- phous carbon EFL, composed essentially of two plasmon excitations with large damping@36#.

A. Aluminum target

First, we study the stopping power of~H2)_n clusters im- pinging on aluminum targets. In Fig. 2 we show the results of S_p, S_intra, and S_interversus cluster velocityv; the behavior of S_p satisfactorily agrees with the compiled experimental data@40–43#.

At low velocities, S_p follows a linear dependence with v, reaches a maximum at v;1.5 a.u., and decreases for large velocities. The behavior of S_intra with velocity is similar to

S_p but its values are roughly half. The velocity dependence of S_inter for a cluster composed of 100 particles predicts a small and negative value for low velocities, but then in- creases and finally dominates the cluster energy loss at large velocities. A similar result was obtained in Ref. @22# but using a d function to describe the ELF of aluminum.

To analyze the size dependence of the cluster stopping power we only study the term S_inter, since S_p and S_intraare independent of the number of atoms in the cluster. In Fig. 3 we present the intermolecular stopping power S_interas a func- tion of cluster size, i.e., the number of particles in the cluster, for various velocities. We emphasize that for cluster veloci- tiesv>2 a.u., we obtain a well-defined maximum in Sinter. This means that for each velocity there is a cluster, to be called ‘‘resonant cluster,’’ for which the energy loss per par- ticle is maximum. This resonant effect takes place when the interference among all the cluster components is constructive and the collective effect is maximum. We note also that the resonant cluster is achieved for intermediate cluster sizes and not for the biggest clusters. A similar outcome has been ob- tained for the stopping of hydrogen clusters in silicon@25# as well as in plasma targets @28#.

The size of the resonant cluster and the corresponding maximum value of S_inter increase with the cluster velocity.

We characterize the resonant cluster configuration by the ra- TABLE I. Parameters used to fit the ELF of aluminum and amorphous carbon through Eq.~6!.

Aluminum Amorphous carbon

i A_i vpl,i~a.u.! gi~a.u.! A_i vpl,i ~a.u.! gi~a.u.!

1 1.1178 0.551 0.035 0.2369 0.23 0.21

2 0.7083 0.945 0.49

FIG. 2. Reduced stopping powers, Sp, Sintra, and Sinter, as a function of the cluster velocity. We present results for a cluster with 100 particles incident on aluminum.

FIG. 3. Reduced intermolecular stoppping power, S_inter, for dif- ferent cluster velocitiesv_r(v_r5v/vF), plotted as a function of the number of particles in the cluster. The target is aluminum, and v_F50.894 a.u.

dius R_cl8 and the maximum value of the intermolecular stop- ping power S_inter8 In Fig. 4 we present the dependence of R_cl8 ^{and S}inter8 with the cluster velocity, respectively. As we can see, the resonant cluster radius follows a linear depen- dence with the cluster velocity. The plasmon pole approxi- mation to describe the target predicts that the maximum in- terference will be when the plasmon wavelength is similar to the cluster radius, that is,

R_cl8>2v vpl

. ~8!

This scale rule, shown as a solid line in Fig. 4, agrees very well with the theoretical data obtained by our model, repre- sented by dots in Fig. 4. This means that, as discussed in Ref.

@22#, the main contribution to the collective effects in the cluster stopping power is due to the interferences between target plasmon oscillations generated by the several particles that compose the cluster.

The maximum value of S_inter divided by the proton stop- ping power S_p is shown also in Fig. 4 as a function of the cluster velocity. The theoretical data follow approximately the relation

S_inter8

S_p >0.8n¯

S

D

and since at high velocities S_p}v²², then S_inter8 goes linearly with the cluster velocity. Therefore, the collective effects be- come relatively more important with increasing velocities.

Similar results were obtained for the energy loss of hydrogen clusters in silicon targets @25# and in plasmas @28#.

B. Amorphous carbon target

Now we analyze the stopping power of~H2)_n clusters in amorphous carbon, as a function of the velocity and cluster size. The ELF for amorphous carbon was modeled here as a sum of two Mermin ELF’s@see Eq. ~6!#, and also for com- parison, using only one Mermin ELF. The two-component model for the ELF provides a more realistic representation of the two resonances observed in the optical spectra of amor- phous carbon @36,37# @see Fig. 1~b!#.

We consider a cluster of hydrogen, composed by 100 par- ticles, bombarding an amorphous carbon target. In Fig. 5 we show the velocity dependence of the relevant terms in the stopping power@see Eq. ~5!#: Sp, S_intra, and S_inter. The solid lines correspond to the results obtained with the two- Mermin-ELF’s model, while the dot-dashed lines represent the results derived from the simpler~but more inaccurate, as we will see below! single-Mermin-ELF model. At low ve- locities, S_p goes linearly with v, reaches a maximum at v;1.5 a.u., and then decreases with velocity. In Ref. @33# it was shown that using a sum of two Mermin ELF’s the S_pof amorphous carbon agrees fairly well with the experimental data. The intramolecular stopping term, S_intra, presents simi- lar trends to S_p, but its values are about 45% of those of S_p in all the range of velocities. The behaviors of S_p and S_intra are similar for both dielectric descriptions of carbon.

The model based in a single Mermin ELF gives more stop- ping than the model using two Mermin ELF’s; this is so because the vpl value used in the single plasmon model is smaller than the high-plasmon peak in the experimental ELF.

However, the results obtained for S_interare very different depending of the model used to describe the target. At low FIG. 4. The left axis represents the radius R_cl8 for the resonant

cluster~at which a maximum in the Sinterappears! as a function of the cluster velocity. Dots, cluster radius for the maximum S_inter, derived from Fig. 3; line, scale rule given by Eq.~8!. The right axis shows the maximum value of the intermolecular stopping power S_inter8 ~normalized to the proton stopping power, Sp) as a function of the velocity. Triangles, values derived from Fig. 3; dashed line, scale rule given by Eq.~9!.

FIG. 5. Velocity dependence of the reduced cluster stopping power: Sp, Sintra, and Sinter. We present data for a cluster with 100 particles, incident on amorphous carbon, whose ELF is modeled by a sum of two Mermin ELF’s~solid line! or a single Mermin ELF

~dot-dashed line!.

cluster velocities we obtain negative values of S_interwhen we use the one-component ELF; this is due to destructive inter- ferences in the stopping power. We have found similar re- sults in calculations for aluminum and silicon targets~whose ELF’s are well described with a single plasmon peak! and it was observed that these negative values become bigger the narrower is the plasmon resonance. However, when we in- clude the two plasma resonances in the ELF the destructive interferences disappear, and we obtain positive values of S_inter in the whole velocity range. In this case, S_inter grows with velocity until it seems to saturate forv.8 a.u., while in the case of a single Mermin ELF we obtain bigger values of S_inter. We find that this apparent saturation corresponds to a smooth maximum, but one should go to still higher velocities to find the subsequent decline.

By comparing the values of the reduced intermolecular and intramolecular stopping terms with the proton stopping power, we find that the intermolecular collective effect is dominant when the cluster velocity satisfies vr>5, where v_r5v/vF, andv_F51.138 a.u. For instance, when vr55 we obtain S_p50.077 a.u., Sintra50.032 a.u. and Sinter50.101 a.u.

for a cluster made of 100 particles.

Also if we compare the values of S_interfor aluminum and amorphous carbon, the conclusion is that the vicinage effects decrease for those materials that present several~and wider!

plasmon resonances in the ELF. These effects will be larger for materials with only one resonance frequency, being more sizeable the longer the mean life of the collective excitations produced in the target.

Now, we evaluate the dependence with the cluster size of the vicinage effect in the stopping power of hydrogen clus- ters impinging in amorphous carbon. We center our study in the intermolecular stopping power S_inter. The reduced inter- molecular stopping power S_inter, as a function of the number of atoms in the cluster, is shown in Fig. 6~a! when the target is described by a sum of two Mermin ELF’s, and in Fig. 6~b!

when a single Mermin ELF was used. We present results for various cluster velocities, 1<vr<6.

From Fig. 6~a!, we observe that, when vr51, Sinter as a function of the number of particles in the cluster shows a saturation ~or a very flat maximum!, starting approximately from ten particles. Similar trends were reported experimen- tally by Ray et al.@15# for the energy loss of hydrogen clus- ters in carbon foils at v;1 a.u.; these experiments found a saturation in the energy loss for small clusters. Unfortu- nately, at present there are no stopping power experiments for high velocities and large cluster sizes. This saturation effect at low velocities could be due to the fact that the projectile velocity is not high enough to excite plasmons, which would produce a coherent interference or resonant be- havior, as we will discuss below. A different behavior arises for cluster velocitiesvr.2. By comparison with the simplest case of aluminum, and following from Eq.~8!, we may ex- pect the appearance of two maxima in the stopping power, associated each with the low- and high-energy plasmons in the amorphous carbon target. Actually, we observe that as a result of this double-plasmon feature~included in the amor- phous carbon ELF!, a plateau is formed for intermediate cluster sizes (N;100). We also find that the position of the second maximum shifts to larger cluster sizes forv_r.4, fall- ing beyond the range covered in this figure.

An attempt to compare our predictions with the experi- mental results from Ref. @15# is not possible, because these experiments were done with small clusters, and our model

~spherical and homogeneous clusters! works better for large than for small clusters; moreover, the structure of the projec- tiles used in the experiments ~a core H31 ion surrounded by neutral H₂ molecules! differs from the structure we have considered.

When using the single-Mermin-ELF model to evaluate the stopping power, Fig. 6~b!, we observe that Sintershows a FIG. 6. S_interversus number of particles in the cluster when the ELF of the amorphous carbon target is described:~a! as sum of two Mermin ELF’s, and~b! by a single Mermin ELF. Different curves are displayed for different cluster velocities, v_r5v/vF, with v_F51.138 a.u.

soft maximum as a function of the number of particles in the cluster, which substantially differs from the previously dis- cussed behavior. Although this maximum is not so pro- nounced as in the case of aluminum ~Fig. 3!, it becomes more evident at higher velocities.

In order to understand these results we present in Fig. 7 the contributions to the value of S_inter arising from each of the two plasma resonances (vpl,150.23 a.u. and vpl,250.945 a.u.! in the ELF of amorphous carbon @cf. Fig.

1~b!# for the case of vr54. The dot-dashed curve in Fig. 6~a!

is the same that the solid curve in Fig. 7, where the dotted line represents the contribution of the high-plasmon (vpl,2) and the dot-dashed line the one due to the low-plasmon (vpl,1). We can observe that the intermolecular stopping term, S_inter, is the superposition of two contributions, due to each of these resonances. In addition, we find that the first peak in S_inter@appearing as a shoulder at N;30–100 in Fig.

6~a! for the case vr54# is due to the high plasma resonance (vpl,250.945 a.u.! in the carbon spectrum of Fig. 1~b!, whereas the largest enhancement effect @second peak, at N;5000, in Fig. 6~a!# is produced by the smaller low-energy plasma resonance (vpl,150.23 a.u.! in Fig. 1~b!. This seems to support the idea that the low plasmon peak in the ELF has an important contribution to the stopping power and, in fact, it becomes dominant for high velocities and large clusters.

Therefore, these results show both the influence of the low energy resonance on the long-range behavior of the dy- namical interactions for large clusters of ions, and the neces- sity of more accurate EFL models taking into account in a

more realistic way the structure of the energy loss peaks determined experimentally.

IV. CONCLUSIONS

The analysis of the energy loss of large clusters of ions in matter, based on the standard dielectric function formalism, but using a more realistic description of the target properties, predicts new and interesting features of the dynamical inter- actions that directly affect the energy dissipated in the me- dium.

We propose an approach in which the target electronic properties, determined from the experimental analysis of the energy-loss functions of aluminum and amorphous carbon targets, are represented by a combination of Mermin-type dielectric functions. In this way, the perturbations produced by the cluster of particles are taken into account in a more accurate way than in previous electron-gas descriptions.

In the case of aluminum, where a well-defined single plas- mon resonance is observed experimentally @35#, our results predict an important resonant behavior in the cluster stop- ping power, which occurs when the size of the cluster is such that most of its particles interfere in a coherent way with the collective perturbation of the medium. The radius of the resonant cluster, for which a maximum stopping enhance- ment is obtained, is proportional to the velocity @Eq. ~8!#, whereas the maximum intermolecular stopping power value, divided by the proton stopping power, is proportional to the third power of the velocity@Eq. ~9!#.

A different behavior is obtained in the case of amorphous carbon, because of the double-plasmon characteristics ob- served in the primary energy-loss spectra@36#. In this case, in the range of high velocities (v_r.3) the contributions from the two broad resonances partially overlap, producing a pla- teau for intermediate cluster sizes, and a wide maximum for very large clusters. Moreover, for intermediate velocities (v_r;1 –2) we find an approximate saturation in the cluster energy loss when the number of particles in the cluster is increased. This behavior is qualitatively in agreement with recent experimental observations of saturation effects in the energy loss of hydrogen clusters in carbon foils@15#.

We emphasize here the importance of using an accurate representation of the energy loss function of each material in order to provide a good description of the interference ef- fects, in particular in the case of swift ion clusters where these effects extend over a long range of distances and be- come sensitive to the structure of the dielectric function.

We think that a realistic study of multiatomic cluster col- lisions and propagation inside a solid should incorporate many more processes than those discussed in this paper:

cluster spread, charge exchange, multiple scattering, and electronic excitations. However, at the present time, some of these effects have not yet been analyzed enough.

ACKNOWLEDGMENTS

Partial support was provided by the Spanish Direccio´n General de Investigacio´n Cientı´fica y Te´cnica, through projects PB92-0341 and PB93-1125. N.R.A. thanks the Con- selleria d’Educacio´ i Cie`ncia de la Generalitat Valenciana for financial support under the program PROPIO.

FIG. 7. Contribution of each plasmon peak in the ELF of amor- phous carbon to the intermolecular stopping power, S_inter, as a function of the number of particles in the cluster. The solid curve,

S_inter, is the sum of the contribution due to the low-plasmon peak

~dot-dashed curve! and the high-plasmon peak ~dotted curve!. The velocity of the cluster relative to the Fermi velocity of amorphous carbon isv_r54. The amorphous carbon target was modeled by a sum of two Mermin ELF’s.

@1# Proceedings of the Conference on Polyatomic Ion Impact on Solids and Related Phenomena, Saint-Malo, 1993, edited by H.

H. Andersen and L. E. Rehn@Nucl. Instrum. Methods B 88, 1

~1994!#.

@2# W. Brandt, A. Ratkowsky, and R. H. Ritchie, Phys. Rev. Lett.

33, 1325~1974!.

@3# M. F. Steuer and R. H. Ritchie, Nucl. Instrum. Methods B 33, 170~1988!; 40/41, 372 ~1989!.

@4# Y. Wang and T. Ma, Phys. Lett. A 178, 209 ~1993!.

@5# A. Bret, Nucl. Instrum. Methods B 88, 107 ~1994!.

@6# J. Jensen, H. H. Mikkelsen, and P. Sigmund, Nucl. Instrum.

Methods B 88, 191~1994!.

@7# M. M. Jakas and N. E. Capuj, Phys. Rev. A 52, 439 ~1995!.

@8# T. Kaneko, Phys. Rev. A 51, 535 ~1995!.

@9# D. S. Gemmell, A. Faibis, E. P. Kanter, Z. Vager, and B. J.

Zabransky, Vacuum 39, 355~1989!.

@10# Y. Susuki, K. Kimura, M. Mannami, N. Sakamoto, H. Ogawa, I. Katayama, T. Noro, and H. Ikegami, Phys. Rev. A 50, 3533

~1994!.

@11# M. Fritz, K. Kimura, Y. Susuki, and M. Mannami, Phys. Rev.

A 50, 2405~1994!.

@12# B. Mazuy, A. Belkacem, M. Chevallier, M. J. Gaillard, J. C.

Poizat, and J. Remillieux, Nucl. Instrum. Methods B 28, 497

~1987!.

@13# B. Mazuy, A. Belkacem, M. Chevallier, M. J. Gaillard, J. C.

Poizat, and J. Remillieux, Nucl. Instrum. Meth. B 33, 105

~1988!.

@14# M. Farizon, A. Clouvas, N. V. Castro Faria, B. Farizon- Mazuy, M.-J. Gaillard, E. Gerlic, A. Denis, J. Desesquelles, and Y. Ouerdane, Phys. Rev. A 43, 121~1991!.

@15# E. Ray, R. Kirsch, H. H. Mikkelsen, J. C. Poizat, and J. Remil- lieux, Nucl. Instrum. Methods B 69, 133~1992!.

@16# K. Baudin, A. Brunelle, M. Chabot, S. Della-Negra, J. Dep- auw, D. Garde`s, P. Ha˚kansson, Y. Le Beyec, A. Billebaud, M.

Fallavier, J. Remillieux, J. C. Poizat, and J. P. Thomas, Nucl.

Instrum. Methods B 94, 341~1994!.

@17# J. C. Eckardt, G. Lantschner, N. R. Arista, and R. A. Bara- giola, J. Phys. C 11, L851~1978!.

@18# R. Levi-Setti, K. Lam, and T. R. Fox, Nucl. Instrum. Methods 194, 281~1982!.

@19# N. R. Arista, Phys. Rev. B 18, 1 ~1978!.

@20# N. R. Arista and A. Gras-Martı´, J. Phys. Condens. Matter 3, 7931~1991!.

@21# I. Abril, M. Vicanek, A. Gras-Martı´, and N. R. Arista, Nucl.

Instrum. Methods B 67, 56~1992!.

@22# M. Vicanek, I. Abril, N. R. Arista, and A. Gras-Martı´, Phys.

Rev. A 46, 5745~1992!.

@23# M. Farizon, N. V. de Castro Faria, B. Farizon-Mazuy, and M.

J. Gaillard, Phys. Rev. A 145, 179~1992!.

@24# E. Nardi and Z. Zinamon, Phys. Rev. A 51, R3407 ~1995!.

@25# F. J. Pe´rez-Pe´rez, I. Abril, N. R. Arista, and R. Garcia-Molina, Nucl. Instrum. Methods B 115, 18~1996!.

@26# E. Nardi, Z. Zinamon, and D. Ben-Hamu, Nuovo Cimento 106, 1839~1993!.

@27# C. Deutsch and N. A. Tahir, Nuovo Cimento 106, 1811 ~1993!.

@28# E. Bringa and N. R. Arista, Phys. Rev. E 52, 3010 ~1995!.

@29# C. Deutsch and N. A. Tahir, Laser Part. Beams 13, 211 ~1995!.

@30# I. F. Silvera, Rev. Mod. Phys. 52, 393 ~1980!.

@31# N. D. Mermin, Phys. Rev. B 1, 2362 ~1970!.

@32# J. Lindhard, Mat. Fys. Medd. Dan. Vid. Selsk. 28, No. 8

~1954!.

@33# I. Abril, R. Garcia-Molina, and N. R. Arista, Nucl. Instrum.

Methods B 90, 72~1994!.

@34# D. J. Planes, R. Garcia-Molina, I. Abril, and N. R. Arista, J.

Electron Spectrosc. Relat. Phenom.~to be published!.

@35# H.-J. Hagemann, E. Gudat, and C. Kunz, J. Opt. Soc. Am. 65, 742 ~1975!; H.-J. Hagemann, E. Gudat, and C. Kunz, Deut- sches Elektronen-Synchrotron Report No. DESY SR-74/7, Hamburg, 1974.

@36# J. Cazaux and D. Gramari, J. Phys. ~Paris! 38, L133 ~1977!.

@37# R. Raether, Excitations of Plasmons and Interband Transitions by Electrons, Springer Tracts in Modern Physics Vol. 88

~Springer, Berlin, 1980!.

@38# M. W. Williams and E. T. Arakawa, J. Appl. Phys. 43, 3460

~1972!.

@39# J. C. Ashley and P. M. Echenique, Phys. Rev. B 16, 8701

~1987!.

@40# H. H. Andersen and J. F. Ziegler, Hydrogen: Stopping Powers and Ranges in All Elements, Vol. 3 of The Stopping and Ranges of Ions in Matter~Pergamon, New York, 1977!.

@41# J. F. Janni, At. Data Nucl. Data Tables 27, 341 ~1982!.

@42# J. F. Ziegler, J. P. Biersack, and U. Littmark, The Stopping and Range of Ions in Solids, Vol. 1 of The Stopping and Ranges of Ions in Matter~Pergamon, New York, 1985!.

@43# International Commission on Radiation Units and Measure- ments Report No. 49, Bethesda, MD, 1993~unpublished!.